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<article class="li"><h3 class="heading">
<span class="type">Item</span><span class="period">.</span>
</h3>
<p><dfn class="terminology">Differentiation in s-Domain</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\cal L}[t f(t)]=-\frac{d}{ds}F(s).
\end{equation*}
</div>
<p class="continuation">This can be proven by differentiating the Laplace transform:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\frac{d}{ds}F(s)=\int_{0}^\infty f(t) \frac{d}{ds} e^{-st} dt
=\int_{0}^\infty (-t) f(t) e^{-st} dt
\end{equation*}
</div>
<p class="continuation">Repeat this process we get</p>
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\begin{equation*}
{\cal L}[t^n f(t)]=(-1)^n\frac{d^n}{ds^n}F(s).
\end{equation*}
</div></article><span class="incontext"><a href="sec8_2.html#li-71" class="internal">in-context</a></span>
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